With the method outlined, we now proceed to the analysis of the results generated by PFE. The analysis is performed for two sample cases: an \(M_j=1.086\)  underexpanded jet issuing from a converging nozzle, and a heated \(M_j=1.35\)  overexpanded jet issuing from a converging-diverging nozzle (design Mach number \(M_d=1.5\) ). The mean flow parameters used in the latter were adjusted to match the conditions from a large eddy simulation (LES). In the first case, the velocity profile proposed by [1]} was used to model the spatial development of the jet. A velocity profile based on [2]} was used in the second case, which was shown to best model the overall behaviour of the mean streamwise velocity. In both cases, the mean temperature profile is obtained using a Crocco-Busemann approximation. The radial coordinate is discretised using Chebyshev polynomials [3]}, with the radial mapping proposed by [4]}; the number of radial collocation points was \(N_r=200\)  for all the cases studied herein. Boundary conditions were implemented as in [4]}, and the maximum radial position of the domain was chosen as \(r_{max}/D=50\) .
